Lemma 2.1 ([1]}, [2]}, [3]})
Let \(1\le p,q,p_1,q_1,p_2,q_2\le \infty \) , with \(\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}\)  and \(\frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2}\) .
Let \(\partial \)  be a partial derivative in space or time, in addition, let \(\partial _t f,\ \nabla f \in L^{p_1}(0,T;L^{q_1}(\mathbb {T}^d))\) , \(g\in L^{p_2}(0,T;L^{q_2}(\mathbb {T}^d))\) . Then, there holds
\(\begin{aligned}&\Vert {\partial (fg)^\varepsilon }-\partial (f\,{g}^\varepsilon )\Vert _{L^p(0,T;L^q(\mathbb {T}^d))}\\\le & C\left(\Vert \partial _{t} f\Vert _{L^{p_{1}}(0,T;L^{q_{1}}(\mathbb {T}^d))}+\Vert \nabla f\Vert _{L^{p_{1}}(0,T;L^{q_{1}}(\mathbb {T}^d))}\right)\Vert g\Vert _{L^{p_{2}}(0,T;L^{q_{2}}(\mathbb {T}^d))}, \end{aligned}\) 
